Stock patterns in a class of delayed discrete-time population models

Rajni; Shuchi Sahu; Suruchi Sarda; Bapan Ghosh

doi:10.3934/dcdss.2024078

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2024. Doi: 10.3934/dcdss.2024078

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Stock patterns in a class of delayed discrete-time population models

1.
Differential Equations, Modeling, and Simulation Group, Department of Mathematics, Indian Institute of Technology Indore, India
2.
Center for Basic Sciences Pt. Ravishankar Shukla University, Raipur, Chhattisgarh, India

^*Corresponding author: Bapan Ghosh
^*Corresponding author: Bapan Ghosh

Received: February 2024

Revised: April 2024

Early access: May 2024

The first author is supported by University Grants Commission (UGC) under the grant 16-9(June2018)/2019(NET/CSIR)

Abstract / Introduction Full Text(HTML) Figure(7) / Table(3) Related Papers Cited by

Abstract

This paper highlights the behavior presented by a new class of discrete-time predator-prey models which have a time delay in the predation process. These models are formulated by discretizing continuous-time delayed Rosenzweig-MacArthur predator-prey systems. The stability analysis is performed through Routh-Hurwitz (RH) criteria, and bifurcation diagrams help to grasp the change in the dynamics of interacting species. First, we examine the impact of changing carrying capacity on (i) the stability of the equilibrium and (ii) the mean population level in delayed systems. When the delay is very large, the stability thresholds of the carrying capacity of different delayed models tend to converge. We compute the mean population size as a function of carrying capacity. Mean prey (resp. predator) stock increases (resp. decreases) with increasing carrying capacity of prey resulting in the paradox of enrichment in our discrete-time systems. For higher delayed models, the value of the mean prey (resp. predator) stock is higher (lower). Second, harvesting has a potential to stabilize an unstable mode of the coexisting equilibrium. When prey is harvested, the mean population of prey (resp. predator) decreases (increases). Therefore, no hydra effect is experienced by prey species. On the other hand, predator harvesting potentially increases its own mean stock, leading to hydra effects in predators. Most importantly, we have computed the mean stock with respect to harvesting. For both prey and predator harvesting, the mean prey size increases with an increase in delay, whereas the mean predator size decreases with an increase in delay. For the system with larger delay, the hydra effects on predators become more prominent in the sense that the rate of increase of mean predator stock is higher.

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Mathematics Subject Classification: Primary: 39A28, 39A33, 65Q10, 92D25.

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Figure 1. Bifurcation diagram with predator population. The bifurcation diagrams of the delayed models with delay $ 0 $, $ 1 $, $ 2 $, $ 3 $, and $ 4 $ units are shown in subplots (b), (c), (d), (e), and (f), respectively. We fixed $ r = 0.4 $

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Figure 2. Phase portrait for (a) K = 5 and (b) K = 6.7. The parameters are the same for the models with delay $ 0 $, $ 1 $, $ 2 $, $ 3 $, and $ 4 $ unit as mentioned in Figure 1. The third subplot (c) represents the phase portrait for five different values of $ K $ for the $ 3 $-unit delayed model

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Figure 3. Mean population for (a) prey and (b) predator corresponding to different delayed models given in Fig. 1, for each value of $ K $. Here, '$ * $' (black colored) represents the equilibrium state. The initial condition used here is $ (x_0, y_0) = (2.1, 1) $, $ u_{i, 0} = x_0 $, and $ v_{i, 0} = y_0 $, where $ i $ depends on the delay value. For any parameter set, we have estimated the mean value by considering $ 300, 000 $ iterations to achieve convergence. The same practice is also followed for the remaining situations

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Figure 4. Bifurcation diagram as a function of $ e_1 $ corresponding to $ 0 $, $ 1 $, $ 2 $, $ 3 $, and $ 4 $ unit delays. The maximum and minimum predator population sizes are plotted when effort is varied

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Figure 5. Mean population for (a) prey and (b) predator corresponding to different delayed models under prey harvesting. Here, '$ * $' (black colored) represents the equilibrium state. The initial condition used here is $ (x_0, y_0) = (2.1, 1) $, $ u_{i, 0} = x_0 $, and $ v_{i, 0} = y_0 $, where $ i $ depends on the delay value

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Figure 6. Bifurcation diagram as a function of $ e_2 $ corresponding to different delayed systems. The bifurcation diagram is plotted by accounting for the maximum and minimum predator sizes in the long run when effort is increased. The initial condition is set as the same for prey harvesting

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Figure 7. Mean population for (a) prey and (b) predator corresponding to different delayed models for each value of $ e_2 $. Here, '$ * $' (black colored) represents the equilibrium state

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Table 1. Computation of threshold value $ K $ corresponding to different delays. We have fixed parameters of the unharvested model as $ \alpha = 3/5 $, $ \beta = 3/10 $, $ h = 1 $, and $ m = 1/5 $

$\mathit{\boldsymbol{ \tau }}$	Threshold $\mathit{\boldsymbol{ K }}$
$\mathit{\boldsymbol{ \tau }}$	$ \mathit{\boldsymbol{r=0.1 }}$	$ \mathit{\boldsymbol{r=0.2 }}$	$ \mathit{\boldsymbol{r=0.3 }}$	$ \mathit{\boldsymbol{r=0.4 }}$	$ \mathit{\boldsymbol{r=0.5 }}$
$ 0 $	$ 4.72727 $	$ 4.72727 $	$ 4.72727 $	$ 4.72727 $	$ 4.72727 $
$ 1 $	$ 4.60275 $	$ 4.60309 $	$ 4.60343 $	$ 4.60377 $	$ 4.60411 $
$ 2 $	$ 4.53515 $	$ 4.53580 $	$ 4.53646 $	$ 4.53711 $	$ 4.53775 $
$ 3 $	$ 4.49440 $	$ 4.49529 $	$ 4.49617 $	$ 4.49704 $	$ 4.49790 $
$ 4 $	$ 4.46806 $	$ 4.46910 $	$ 4.47013 $	$ 4.47115 $	$ 4.47215 $

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Table 2. Stability thresholds $e_1$ of the models when delay is $0$, $1$, $2$, $3$, and $4$ units. In the absence of predator harvesting (i.e., $q_2 = 0$), we select the ecological parameters of the models as $r = 0.4$, $K = 6$, $\alpha = 6/10$, $\beta = 3/10$, $h = 1$, and $m = 0.2$. The technological parameter $q_1 = 1$ is chosen for simulation

$\tau$	0	1	2	3	4
Threshold $e_1$	$0.08484$	$0.09310$	$0.09756$	$0.10025$	$0.10199$

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Table 3. Stability threshold $ e_2 $ corresponding to different time delays. The fixed ecological parameters of the model are $ r = 0.4 $, $ K = 6 $, $ \alpha = 6/10 $, $ \beta = 3/10 $, $ h = 1 $, and $ m = 0.2 $, along with the technological parameter $ q_2 = 1. $

$ \tau $	0	1	2	3	4
Threshold $ e_2 $	$ 0.01755 $	$ 0.01890 $	$ 0.01959 $	$ 0.01998 $	$ 0.02023 $

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